Fig. 5

Phase diagram of phosphorene, calculated by a Monte Carlo simulation for the 2D Ising model with nearest and next-nearest neighbor interactions on a \(32 \times 32\) lattice, averaged over 20 phase diagrams. The high-temperature phase transition is calculated from the peaks in the specific heat. To improve accuracy, we also tracked secondary observables such as magnetization and domain length to better distinct the phases. The error bars on the phase boundaries represent statistical uncertainty derived from Monte Carlo sampling, as well as thermal fluctuations near critical temperatures. There is another low-temperature phase transition that only obtained from using quenched Monte Carlo simulation and is not observed in the Monte Carlo with annealing, as explained in the Fig. 4. The exact boundary in the low-temperature glassy state phase transition is obtained from change in the magnetization and also an analysis on the spin overlap parameter, described by Eq. (5). There are two orders observed at different values parameter \(\alpha = J/J^{\prime }\). At \(\alpha < 2\), there is a new ordered state that opposite double stripes repeat alternately in the horizontal direction, which called it as checker-stripy order. At \(\alpha > 2\), the order is the usual antiferromagnetic order, so called checkerboard order, in which each spin is surrounded by spins in opposite directions. Both orders were obtained well away from the transition points, where thermal noise is minimal. Error bars were also calculated to account for the uncertainty arising from thermal fluctuations when approaching a phase transition point.